Presentation on theme: "MSP15 The Fourier Transform (cont’) Lim, 1990. MSP16 The Fourier Series Expansion Suppose g(t) is a transient function that is zero outside the interval."— Presentation transcript:

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MSP16 The Fourier Series Expansion Suppose g(t) is a transient function that is zero outside the interval [-T/2,T/2] (e.g., a cycle of a periodic function). We can obtain a sequence of coefficients by making s a discrete variable and integrating over the interval (with period T), so that

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MSP18 The Discrete Fourier Transform (DFT) If we discretize both time and frequency the Fourier transform pair of a series become

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MSP19 The DFT (cont’) If {f i } is a sequence of length N (by taking samples of a continuous function at equal intervals) then its discrete Fourier transform pair is given by

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MSP20 Properties of the Fourier Transform The addition theorem (addition in time/spatial domain corresponds to addition in frequency) The shift theorem (shifting a function causes to only phase shift) The convolution theorem (convolution is equivalent to multiplication in the other domain) …